N-body Systems with Linear Wind Resistance: Analytic Solutions

The attractive and repulsive linear Hookean form of gravity is known to allow for analytic solutions to N-body systems for arbitrary masses and initial conditions. This linear system is very well suited for use in the advanced undergraduate classroom with the position of every particle rj(t) known analytically in terms of only sine, cosine, and exponential functions. Here it is shown that simple analytic solutions are also possible when linear wind resistance is included, provided that the force due to wind resistance force on each particle is proportional to the mass of that particle. Surprising analytic expressions for the behavior of the system if only a single particle in the system is subject to a linear wind resistance are also presented.